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9.57
Theorem.
Let be real numbers. If is a function that is integrable on each
interval
with endpoints in then
Proof: The case where
is proved in
theorem 8.18. The rest of
the proof is
exactly like the proof of exercise 5.69.
9.58
Exercise.
Prove theorem
9.57.
We have proved the following formulas:
In each case we have a formula of the form
This is a general sort of situation, as is shown by the following theorem.
Proof: Choose a point and define
Then for any points in we have
We've used the fact that
9.62
Definition (Indefinite integral.)
Let
be a function that is integrable on
every subinterval
of an interval
. An
indefinite integral for on is any
function
such that
for all
.
A function that has an indefinite integral always has infinitely many
indefinite integrals,
since if is an indefinite integral for then so is
for any number :
The following notation is used for indefinite integrals. One writes to denote
an indefinite integral for . The here is a dummy variable and can be
replaced by any
available symbol. Thus, based on formulas (9.59) - (9.60),
we
write
We might also write
Some books always include an arbitrary constant with indefinite integrals,
e.g.,
The notation for indefinite integrals is treacherous. If you see the
two equations
and
then you want to conclude
|
(9.63) |
which is wrong.
It would be more logical to let the symbol denote the set
of all indefinite integrals for . If you see the statements
and
you are not tempted to make the conclusion in (9.63).
Proof:
The statement (9.65) means that if is an indefinite integral
for and is an indefinite integral for , then
is an indefinite integral for .
Let be an indefinite integral for and let be an indefinite integral
for .
Then for all
It follows that is an indefinite integral for .
9.66
Notation (.)
If
is a function defined on an interval
,
and if
are
points in
we write
for
. The
here is a dummy
variable,
and sometimes the notation is ambiguous, e.g.
. In such cases
we may write
. Thus
while
Sometimes we write
instead of
.
9.67
Example.
It follows from our notation that if
is an indefinite integral for
on an
interval
then
and this notation is used as follows:
In the last example I have implicitly used
9.68
Example.
By using the trigonometric identities from theorem
9.21 we can
calculate
integrals of the form
where
are
non-negative integers
and
. We will find
We have
so
Thus
Hence
The method here is clear, but a lot of writing is involved, and there are many
opportunities
to make errors. In practice I wouldn't do a calculation of this sort by hand.
The Maple
command
> int((sin(x))^3*cos(3*x),x=0..Pi/2);
responds with the value
- 5/12
9.69
Exercise.
A
Calculate the integrals
Then determine the values of
without doing any calculations. (But include an explanation of where your
answer comes
from.)
9.71
Exercise.
Arrange the numbers
in increasing order. Try to do the problem without
making any
explicit calculations. By making rough sketches of the graphs you should be
able to come up
with the answers. Sketch the graphs, and explain how you arrived at your
conclusion. No
`` proof" is needed.
Next: 10. Definition of the
Up: 9. Trigonometric Functions
Previous: 9.3 Integrals of the
  Index
Ray Mayer
2007-09-07